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e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 | import OAI.Combinatorics.ExpectationThreshold.ProbabilityFinite
import Mathlib.Tactic.NormNum
namespace OAI
/-!
# The numerical contradiction used in rounding
A nonnegative score bounded below by one quarter on an event of probability
at least nine tenths has expectation at least nine fortieths.
-/
namespace TalagrandThreshold
variable {Ω : Type*} [Fintype Ω]
theorem expectation_ge_nine_fortieths (μ : FiniteLaw Ω) (P : Ω → Prop)
(Z : Ω → ℝ) (hZ : ∀ ω, 0 ≤ Z ω) (hprob : (9 / 10 : ℝ) ≤ μ.prob P)
(hlower : ∀ ω, P ω → (1 / 4 : ℝ) ≤ Z ω) :
(9 / 40 : ℝ) ≤ μ.expectation Z := by
have hinclusion : μ.prob P ≤ μ.prob (fun ω => (1 / 4 : ℝ) ≤ Z ω) :=
μ.prob_mono hlower
have hmarkov := μ.prob_ge_le_expectation_div Z hZ
(by norm_num : (0 : ℝ) < 1 / 4)
have hbound : (9 / 10 : ℝ) ≤ μ.expectation Z / (1 / 4 : ℝ) :=
hprob.trans (hinclusion.trans hmarkov)
calc
(9 / 40 : ℝ) = (9 / 10 : ℝ) * (1 / 4 : ℝ) := by norm_num
_ ≤ μ.expectation Z := (le_div_iff₀ (by norm_num : (0 : ℝ) < 1 / 4)).mp hbound
theorem rounding_expectation_contradiction (μ : FiniteLaw Ω) (P : Ω → Prop)
(Z : Ω → ℝ) (hZ : ∀ ω, 0 ≤ Z ω) (hprob : (9 / 10 : ℝ) ≤ μ.prob P)
(hlower : ∀ ω, P ω → (1 / 4 : ℝ) ≤ Z ω)
(hupper : μ.expectation Z ≤ (1 / 10 : ℝ)) : False := by
have h := (expectation_ge_nine_fortieths μ P Z hZ hprob hlower).trans hupper
exact (by norm_num : ¬ ((9 / 40 : ℝ) ≤ 1 / 10)) h
end TalagrandThreshold
end OAI
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